In "my collection", you have the option to sell a card to the public market, private or to kate (less)
When you are trading with someone, you must send the cards to his private market.If you want to sell a card but you don't know who wants it, sell it on the public market and someone will come and buy it (depending on the card and the price, it can take some time).
Selling your cards to kate will be fast but will give you less money.
At this time there is no function to trade cards with other players other than using the Market (selling cards for clintz on the open market) or selling via a private sale (only to a certain person).
However, you need to buy credits from the Shop before you can sell cards I do believe. Although this does not have any effect on you buying cards from the Market using clintz you have earned.